Characterizing linear groups in terms of growth properties

Linear growth for semigroups which are disjoint unions of finitely many copies of the free monogenic semigroup

We show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has linear growth. This implies that the the corresponding semigroup algebra is a PI algebra.

关于线性群的共轭增长

On conjugacy growth of linear groups

本文研究了非线性生成的线性群的共轭增长性。我们证明了GL_d的非虚可解子群具有一致的指数共轭增长，并且实际上，对于字度量，作为半径为n的球的元素的特征多项式而产生的不同多项式的数目仅以维数d有界地从0以指数增长性增长.

通过循环群刻画层次超几何自由群

Characterizing hierarchically hyperbolic free by cyclic groups

本文通过研究具有粗中位的自由群，证明其等价于层次超几何群和CAT(0)立方复形的准等距性，提出'无分支块'条件和'过度线性'性质。

具有中间增长的线性半群族

Families of linear semigroups with intermediate growth

利用加性数论中的方法构造了具有中间增长性的非线性生成线性半群族。

On groups with linear sci growth

We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented groups we show that the linear sci is preserved under amalgamated products over finitely generated one-ended groups. Eventually one proves that most non-uniform lattices have linear sci.

Horofunctions on graphs of linear growth

We prove that a linear growth graph has finitely many horofunctions. This provides a short and simple proof that any finitely generated infinite group of linear growth is virtually cyclic.

Harmonic functions of linear growth on solvable groups

In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner's theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner's proof for Gromov's theorem on groups of polynomial growth.